Show that the function is a joint density function and find the required probability. f(x, y) 0
Show that the function is a joint density function and find the required probability. f(x, y) 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![A joint density function of the continuous random variables \(x\) and \(y\) is a function \(f(x, y)\) satisfying the following properties.
a. \( f(x, y) \geq 0 \) for all \((x, y)\)
b. \[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dA = 1
\]
c. \[
P[(x, y) \in R] = \int_R f(x, y) \, dA
\]
Show that the function is a joint density function and find the required probability.
\[
f(x, y) =
\begin{cases}
\frac{1}{4}, & 0 \leq x \leq 1, \ 1 \leq y \leq 5 \\
0, & \text{elsewhere}
\end{cases}
\]
\[
P(0 \leq x \leq 1, \ 1 \leq y \leq 2)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa836ac4e-4db0-475d-87c6-ece7bcb44ff3%2Ffafd2597-f651-4b8d-b1eb-3518264f40be%2Fynj5s3r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A joint density function of the continuous random variables \(x\) and \(y\) is a function \(f(x, y)\) satisfying the following properties.
a. \( f(x, y) \geq 0 \) for all \((x, y)\)
b. \[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dA = 1
\]
c. \[
P[(x, y) \in R] = \int_R f(x, y) \, dA
\]
Show that the function is a joint density function and find the required probability.
\[
f(x, y) =
\begin{cases}
\frac{1}{4}, & 0 \leq x \leq 1, \ 1 \leq y \leq 5 \\
0, & \text{elsewhere}
\end{cases}
\]
\[
P(0 \leq x \leq 1, \ 1 \leq y \leq 2)
\]
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